# purely inseparable

Let $F$ be a field of characteristic^{} $p>0$ and let $\alpha $ be an element^{} which is algebraic over $F$. Then $\alpha $ is said to be *purely inseparable* over $F$ if ${\alpha}^{{p}^{n}}\in F$ for some $n\ge 0$.

An algebraic field extension $K/F$ is *purely inseparable* if each element of $K$ is purely inseparable over $F$.

Purely inseparable extensions have the following property: if $K/F$ is purely inseparable, and $A$ is an algebraic closure^{} of $F$ which contains $K$, then any homomorphism^{} $K\to A$ which fixes $F$ necessarily fixes $K$.

Let $K/F$ be an arbitrary algebraic extension. Then there is an intermediate field $E$ such that $K/E$ is purely inseparable, and $E/F$ is separable^{}.

###### Example.

Let $s$ be an indeterminate, and let $K={\mathbb{F}}_{3}(s)$ where ${\mathbb{F}}_{3}$ is the finite field^{} with $3$ elements. Let $F={\mathbb{F}}_{3}({s}^{6})$. Then $K/F$ is neither separable, nor purely inseparable. Let $E={\mathbb{F}}_{3}({s}^{3})$. Then $E/F$ is separable, and $K/E$ is purely inseparable.

Title | purely inseparable |
---|---|

Canonical name | PurelyInseparable |

Date of creation | 2013-03-22 14:49:08 |

Last modified on | 2013-03-22 14:49:08 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 6 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 12F15 |